Slab (geometry)

In geometry, a slab is a region between two parallel lines in the Euclidean plane,^[1] or between two parallel planes in three-dimensional Euclidean space or between two hyperplanes in higher dimensions.^[2]

Set definition

A slab can also be defined as a set of points:^[3] $\{x\in \mathbb {R} ^{n}\mid \alpha \leq n\cdot x\leq \beta \},$ where $n$ is the normal vector of the planes $n\cdot x=\alpha$ and $n\cdot x=\beta$ .

Or, if the slab is centered around the origin:^[4] $\{x\in \mathbb {R} ^{n}\mid |n\cdot x|\leq \theta /2\},$ where $\theta =|\alpha -\beta |$ is the thickness of the slab.

References

^ Preparata, Franco P.; Shamos, Michael Ian (1985). "2.2.2.1 The slab method". Computational Geometry: An Introduction. New York: Springer. pp. 45–48. doi:10.1007/978-1-4612-1098-6. ISBN 978-1-4612-7010-2. S2CID 206656565.
^ Jacob, Goodman. "Handbook of Discrete and Computational Geometry". CRC Press LLC. Retrieved 24 July 2022.
^ S., Boyd. "Convex Optimization". Cambridge University Press. Retrieved 14 March 2022.
^ Jean-Luc, Marichal; Mossinghoff, Michael J. (2008). "Slices, slabs, and sections of the unit hypercube" (PDF). Online Journal of Analytic Combinatorics. 3 (1). arXiv:math/0607715.

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[Preparata-Shamos-1] Preparata, Franco P.; Shamos, Michael Ian (1985). "2.2.2.1 The slab method". Computational Geometry: An Introduction. New York: Springer. pp. 45–48. doi:10.1007/978-1-4612-1098-6. ISBN 978-1-4612-7010-2. S2CID 206656565.

[Goodman-2] Jacob, Goodman. "Handbook of Discrete and Computational Geometry". CRC Press LLC. Retrieved 24 July 2022.

[Boyd-3] S., Boyd. "Convex Optimization". Cambridge University Press. Retrieved 14 March 2022.

[Marichal-4] Jean-Luc, Marichal; Mossinghoff, Michael J. (2008). "Slices, slabs, and sections of the unit hypercube" (PDF). Online Journal of Analytic Combinatorics. 3 (1). arXiv:math/0607715.

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Slab (geometry)

Set definition

See also

References